AI 中文总结
研究平衡 4 部 4 图中完美匹配的最小顶点度阈值，通过\(k\)部\(k\)图的归约框架，将完美分数匹配存在性转化为有限维优化问题来求解，确定了相应阈值。
AI 中文摘要
一个平衡的\(k\)部\(k\)图是一种\(k\)均匀超图，其每条边与每个划分类恰好交于一个顶点，每个划分类大小为\(n\)。Lo 和 Markström（2014 年）确定了平衡 3 部 3 图中完美匹配的最小顶点度阈值。本文确定了平衡 4 部 4 图的最小顶点度阈值。证明依赖于\(k\)部\(k\)图的归约框架，通过该框架将完美分数匹配的存在性转化为有限维优化问题。
英文摘要
A balanced $k$-partite $k$-graph is a $k$-uniform hypergraph such that every edge intersects each partition class in exactly one vertex, where each partition class has size $n$. Lo and Markström (2014) determined the minimum vertex-degree threshold for perfect matchings in balanced \(3\)-partite \(3\)-graphs. In this paper, we determine the minimum vertex-degree threshold for balanced \(4\)-partite \(4\)-graphs. The proof relies on a reduction framework for \(k\)-partite \(k\)-graphs, through which the existence of a perfect fractional matching is converted into a finite-dimensional optimization problem.
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